This is why I don't like mixed fractions. Simply put, in line 2 you added 5/8 when you should have subtracted it.

There are two ways to handle this. The first is to simply write the difference in terms of whole numbers and fractions, which is what you were trying to do. It reads like this:
(63 + 1/8) - (59 + 5/8) = (63 - 59) + (1/8 - 5/8) = 4 - 1/2 = 3 + 1/2

The other way is to write the whole thing in terms of fractions. It's a bit trickier but works a lot better.
which again works out to be 3 + 1/2.